We prove that if \(T\) is a continuous linear functional on the space \({\mathcal D}\) of Henstock-Kurzweil integrable functions on \([a_1,b_1] \times \cdots \times [a_m,b_m]\), then there exists a function \(g\) of strong bounded variation on \([a_1,b_1] \times \cdots \times [a_m,b_m]\) such that \[ T(f) = (HK) {\int \ldots \int \atop \scriptstyle{[a_1,b_1] \times \cdots \times [a_m,b_m]}} f(x_1,\ldots,x_m) g(x_1,\ldots,x_m)\, dx_1 \ldots dx_m\, . \]

In this paper we show that there are close relationships amongst Cantor bar totality, non-integer integral and Hausdorff integral. If a singular function \(f\) has zero Lipschitz (\(1-\nu\))-numbers on a Cantor set \(C\) with \(H\)-dim~\(C=1-\nu\), \(0\lt \nu\lt 1\), then the \(1-\nu\) order fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Moreover, under strong assumptions on the function \(f\) the \(1-\nu\) order derivative of \(f\) exists everywhere. Consequently, the \(\nu\) order fractional integral of \(f\) equals \(f(0)t^{\nu}\). Using the Concept of th Hausdorff derivative we prove that if a singular function has a Hausdorff derivative on \(C\), then the fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Finally, under some assumptions on \(f\) and \(C\), we establish \begin{align*} (D^{-\nu}f)(t)=&\int_0^t\int_0^uv^{\nu-1}f'_H(u-v)\,dv^{1-\nu}\,du\\ =&\int_0^tu^{1-\nu}J_{\nu}(v^{\nu-1}f'_H(u-v))(u)\,du \end{align*} This identity is similar to the identity \[(D^{-\nu}f)(t)=\frac{1}{B_{\nu}}t^{\nu}(J_{\nu}f)(t)\] which professor R. R. Nigmatullin claimed.

This note answers the question, for which Hausdorff functions \(h\) may the \(h\)-Hausdorff and \(h\)-packing measures agree on some subset \(A\) of \(\mathbb{R}^n\), and be positive and finite. We show that these conditions imply that \(h\) is a regular density function, in the sense of Preiss, and that for each such function there is a subset of \(\mathbb{R}^n\) on which the \(h\)-Hausdorff and \(h\)-packing measures agree and are positive and finite.

Classical theorems due to Erdös concerning the measure of sumsets and ejective sets are considered.

It is well known that the derivative of a function of one variable has the Darboux property. In this paper it is shown that the gradient of a differentiable function of several variables maps certain closed convex sets to connected sets.

Theorem. If Henstock integrable \(f_n\) converge in measure to a finite \(f\) and their primitives \(F_n\) are equi-\(ACG_*\) and converge pointwise to a continuous \(F\) then \(\int f = \lim F_{n}\).

We discuss a question on the existence of partial \(\mu\)-nonmeasurable \(H\)-selectors, where \(\mu\) is a given nonzero \(\sigma\)-finite measure defined on some \(\sigma\)-algebra of subsets of a set \(E\) and quasiinvariant under an uncountable group \(G\) of transformations of \(E\), and \(H\) is an arbitrary countable subgroup of \(G\).

It is well known than an ordinary (trigonometric) Fourier series “can” be integrated term by term; that is, whether the Fourier series of an integrable function \(f\) is convergent or not, the series obtained by term by term integration of it is convergent to the corresponding integral of \(f\). There is also a more general theorem in which \(f\) and its Fourier series are first multiplied by a function of bounded variation before integrating. This paper aims to obtain similar theorems for the Fourier-Legendre series discussed by Love and Hunter in Proc. London Math. Soc (3) 64 (1992) 579-601 and by Love in that journal (3) 69 (1994) 629-672. Most of the paper is occupied by lemmas which lead to establishing the dominated convergence of the partial sums of the Fourier-Legendre series under certain conditions. For several reasons one of these conditions is that the interval of integration must be a closed subinterval of \((-1,1)\), the interval on which the Legendre functions \(P_\nu^\mu (x)\) are defined; further, I have only had success with integration over subintervals of \([- \frac{1}{2} \sqrt{3}, \frac{1}{2}\sqrt{3}]\).

Recent research has shown that there is a significant cleavage between the structure of ω sets for continuous functions, and the structure of ω-limit sets for Lipschitz functions. While every non-empty nowhere dense compact set is an ω-limit set for a continuous function, most of these sets cannot be an attractor for a Lipschitz function. When one considers the topological structure of these two classes of ω-limit sets, however, there is no such divergence. In this paper we investigate the necessarily measure based structural differences between these two classes of sets. We then use these results to work towards a characterization of ω-limit sets for Lipschitz functions.

It is proved that \(g(C\times C)\) contains an interval if \(g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) satisfies appropriate conditions and \(C\) is the Cantor set.

C. E. Weil formulated the following problem: “Assume that \(f\) is a differentiable real-valued function of \(N\) real variables, \(N \geq 2\), and let \(g = \nabla f\) denote its gradient, which is a function from \(^N\) to \(\mathbb{R}^N\). Let \(G \subset \mathbb{R}^N\) be a nonempty open set and let \(g^{-1}(G) \ne \emptyset.\) Does \(g^{-1}(G)\) have positive \(N\)-dimensional Lebesgue measure?” For \(N = 1\) the answer is yes as was first proved by Denjoy in 1916. Z. Buczolich gave a partial answer to this problem showing that \(g^{-1}(G)\) has positive one-dimensional Hausdorff measure. In other words, he proved that the gradient has the “one-dimensional Denjoy-Clarkson property”. In the present article, we prove the Buczolich result using a quite different method. Moreover, our method gives improvements and generalizations.

Suppose that \(a \lt b\) and each of \(h\) and \(m\) is a real-valued function defined on \([a;b]\) with \(m\) nondecreasing such that if \([p;q]\subseteq [a;b]\) and \(m\vert_p^q = 0\), then \(h\vert_p^q = 0\). There are developed, among other things, necessary and sufficient conditions in order that for each real-valued function \(f\) defined and quasi-continuous on \([a;b]\), the Hellinger-type integral \[\int_{[a;b]}{{dfdh}\over {dm}}\] exists. As is well known, this integral has arisen in connection with, among other things, representation theorems for certain classes of continuous linear functionals.

It is shown that for any \(r\in (0,1)\) and for any continuous function from the unit interval to itself, there are sets of arbitrarily small Lebesgue measure whose preimage has arbitrarily large \(r\)-Hausdorff capacity. This is generalized to functions from the unit square to the interval.

We answer two questions asked by Rosen in 1994. We show that the class of extendable connectivity functions from \(I\) into \(I\) (\(I = [0,1]\)) cannot be characterized in terms of associated sets, and we show that one of Jones’ functions obeying \(f(x+y) = f(x) + f(y)\) is an example of an almost continuous function from \(\mathbb{R}\) into \(\mathbb{R}\) which is not the uniform limit of any sequence of extendable connectivity functions.

The main result: For every family \(\mathcal{G}\) of additive functions with \(\text{card }{\mathcal{G}}=2^\omega\) if the covering of the family of all level sets of functions from \(\mathcal{G}\) is equal to \(2^\omega\), then there exists an additive Darboux function \(f\) such that \(f+g\) is Darboux for no \(g\in\mathcal{G}\).

Conditions (in terms of integrals of the weights) are derived, under which the weighted \(L^q\)-norm of the \(j\)-th order derivative of the function \(u\) can be estimated by the sum of the weighted \(L^r\)-norm of \(u\) and of the weighted \(L^p\)-norm of its \(m\)-th order derivative, \(j\lt m\). All mutual positions of the parameters, \(p,\, q, \, r\) are admissible.

We prove that the \(n\)-dimensional Perron integral with respect to the full interval basis, without any regularity condition, defined by continuous major and minor functions is equivalent to the one defined by major and minor functions which are not supposed to be continuous.

It is shown that each real cliquish function \(f\) defined on a pseudometrizable space is the sum of two quasicontinuous functions. If moreover \(f\) is bounded (in the Baire class \(\alpha\)), then we can take the summands with this property.

The paper solves the question whether the limit of transfinite convergent sequence of derivatives is again the derivative. It shows that this problem cannot be solved in the Zermelo-Fraenkel axiomatic system and that this statement is equivalent to the covering number for Lebesgue null ideal being bigger that \(\aleph_1\). In the second part of the paper author proved an analogue of Preiss’s theorem [P] for the transfinite sequences of derivatives.

Given any countable collection of Baire one functions there is a single trajectory from which each of the functions is first return recoverable.

We define a concept of generalized absolute continuity for additive functions of figures. This makes it possible to give a descriptive definition of the ℱ-integral introduced by Pfeffer in 1993. Finally, we discuss a possible extension to additive functions of sets of bounded variation.

In this paper the authors define a multifunction \(F:X \mapsto Y\) to be upper (respectively, lower) \(\beta\)-continuous if \(F^+(V)\) (resp. \(F^-(V))\) is \(\beta\)-open in \(X\) for every open set \(V\) of \(Y\). They obtain some characterizations and several properties concerning upper (resp. lower) \(\beta\)-continuous multifunctions. The relationships between these multifunctions and quasi continuous multifunctions are investigated.

There are two different definitions about the approximately continuous Henstock integral or \(AH\) integral, for short. In this paper we prove that they are equivalent, and that the \(AH\) integral is included in the \(AD\) integral defined by Kubota in 1964. Also we give a solution for the problem put forward by Gordon (1994, p. 255).

In this paper, we give a necessary and sufficient condition in terms of Lebesgue integrable functions for Henstock integrability in Euclidean space.

The change of variables formula for the Riemann integral is discussed and a theorem is proved which perhaps compares favorably with its counterpart in Lebesgue theory.

This note deals with certain recent results concerning summability \(|\overline{N},p_n|_k\).

In this paper the author constructs a \(C^1\) function \(G\) and a Riemann integrable function \(H\), and shows that the composition \(H \circ G\) is not Riemann integrable.

We show that if \(X\) has the property that every continuous image into Baire space is bounded and \(2^\omega\) is not a continuous image of \(X\), then \(X\) is always of first category in some additive sense. This gives an answer to an oral question of L. Bukovský, whether every wQN set has the latter property.

We study \(L^p\) to \(L^q\) mapping properties of nonconvolution singular integral operators on \(\mathbb{R}^1\), whose kernels are obtained by truncating the Hilbert kernel \(1/x\) in ways that depend linearly on the input variable. Some of these operators arise as special cases of the bilinear Hilbert transform and they are shown to map \(L^p\) to \(L^q\) for \(q \lt p\).

We construct a symmetrically continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) such that for some \(X\subset\mathbb{R}\) of cardinality continuum \(f|X\) is of Sierpiński-Zygmund type. In particular such an \(f\) is not countably continuous. This gives an answer to a question of Lee Larson.

A valid proof that Kubota’s AD-integral is more general than Burkill’s AP integral is given.

Two problems related to the characterization of continuity are discussed. In the first problem “\(f\) is almost continuous in the sense of Stallings” will be replaced with a weaker condition “\(f\) is a Darboux function” and it will be shown that the characterization of continuity remains true. Also it follows that for the classes of functions considered, “\(f\) is a Darboux function” is the weakest possible condition for which the characterization remains true. In the second problem “\(f\) is almost continuous in the sense of Stallings” will be replaced with a stronger condition “\(f\) is an extendable function”. Then it will be shown that this condition and conditions (2) and (3) are not redundant.